3.123 \(\int x^m \cos ^4(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=266 \[ \frac {(m+1) x^{m+1} \cos ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+(m+1)^2}+\frac {12 b^2 (m+1) n^2 x^{m+1} \cos ^2\left (a+b \log \left (c x^n\right )\right )}{\left (4 b^2 n^2+(m+1)^2\right ) \left (16 b^2 n^2+(m+1)^2\right )}+\frac {4 b n x^{m+1} \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^3\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+(m+1)^2}+\frac {24 b^3 n^3 x^{m+1} \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{\left (4 b^2 n^2+(m+1)^2\right ) \left (16 b^2 n^2+(m+1)^2\right )}+\frac {24 b^4 n^4 x^{m+1}}{(m+1) \left (4 b^2 n^2+(m+1)^2\right ) \left (16 b^2 n^2+(m+1)^2\right )} \]

[Out]

24*b^4*n^4*x^(1+m)/(1+m)/((1+m)^2+4*b^2*n^2)/((1+m)^2+16*b^2*n^2)+12*b^2*(1+m)*n^2*x^(1+m)*cos(a+b*ln(c*x^n))^
2/((1+m)^2+4*b^2*n^2)/((1+m)^2+16*b^2*n^2)+(1+m)*x^(1+m)*cos(a+b*ln(c*x^n))^4/((1+m)^2+16*b^2*n^2)+24*b^3*n^3*
x^(1+m)*cos(a+b*ln(c*x^n))*sin(a+b*ln(c*x^n))/((1+m)^2+4*b^2*n^2)/((1+m)^2+16*b^2*n^2)+4*b*n*x^(1+m)*cos(a+b*l
n(c*x^n))^3*sin(a+b*ln(c*x^n))/((1+m)^2+16*b^2*n^2)

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Rubi [A]  time = 0.12, antiderivative size = 260, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4488, 30} \[ \frac {(m+1) x^{m+1} \cos ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+(m+1)^2}+\frac {12 b^2 (m+1) n^2 x^{m+1} \cos ^2\left (a+b \log \left (c x^n\right )\right )}{20 b^2 (m+1)^2 n^2+64 b^4 n^4+(m+1)^4}+\frac {4 b n x^{m+1} \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^3\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+(m+1)^2}+\frac {24 b^3 n^3 x^{m+1} \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{20 b^2 (m+1)^2 n^2+64 b^4 n^4+(m+1)^4}+\frac {24 b^4 n^4 x^{m+1}}{(m+1) \left (4 b^2 n^2+(m+1)^2\right ) \left (16 b^2 n^2+(m+1)^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Cos[a + b*Log[c*x^n]]^4,x]

[Out]

(24*b^4*n^4*x^(1 + m))/((1 + m)*((1 + m)^2 + 4*b^2*n^2)*((1 + m)^2 + 16*b^2*n^2)) + (12*b^2*(1 + m)*n^2*x^(1 +
m)*Cos[a + b*Log[c*x^n]]^2)/((1 + m)^4 + 20*b^2*(1 + m)^2*n^2 + 64*b^4*n^4) + ((1 + m)*x^(1 + m)*Cos[a + b*Lo
g[c*x^n]]^4)/((1 + m)^2 + 16*b^2*n^2) + (24*b^3*n^3*x^(1 + m)*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/((1
+ m)^4 + 20*b^2*(1 + m)^2*n^2 + 64*b^4*n^4) + (4*b*n*x^(1 + m)*Cos[a + b*Log[c*x^n]]^3*Sin[a + b*Log[c*x^n]])
/((1 + m)^2 + 16*b^2*n^2)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 4488

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[((m + 1)*(e*x)
^(m + 1)*Cos[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2), x] + (Dist[(b^2*d^2*n^2*p*(p - 1))/(b
^2*d^2*n^2*p^2 + (m + 1)^2), Int[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^(p - 2), x], x] + Simp[(b*d*n*p*(e*x)^(m +
1)*Sin[d*(a + b*Log[c*x^n])]*Cos[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2), x]) /; Free
Q[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + (m + 1)^2, 0]

Rubi steps

\begin {align*} \int x^m \cos ^4\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {(1+m) x^{1+m} \cos ^4\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+16 b^2 n^2}+\frac {4 b n x^{1+m} \cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+16 b^2 n^2}+\frac {\left (12 b^2 n^2\right ) \int x^m \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx}{(1+m)^2+16 b^2 n^2}\\ &=\frac {12 b^2 (1+m) n^2 x^{1+m} \cos ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^4+20 b^2 (1+m)^2 n^2+64 b^4 n^4}+\frac {(1+m) x^{1+m} \cos ^4\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+16 b^2 n^2}+\frac {24 b^3 n^3 x^{1+m} \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{(1+m)^4+20 b^2 (1+m)^2 n^2+64 b^4 n^4}+\frac {4 b n x^{1+m} \cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+16 b^2 n^2}+\frac {\left (24 b^4 n^4\right ) \int x^m \, dx}{(1+m)^4+20 b^2 (1+m)^2 n^2+64 b^4 n^4}\\ &=\frac {24 b^4 n^4 x^{1+m}}{(1+m) \left ((1+m)^4+20 b^2 (1+m)^2 n^2+64 b^4 n^4\right )}+\frac {12 b^2 (1+m) n^2 x^{1+m} \cos ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^4+20 b^2 (1+m)^2 n^2+64 b^4 n^4}+\frac {(1+m) x^{1+m} \cos ^4\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+16 b^2 n^2}+\frac {24 b^3 n^3 x^{1+m} \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{(1+m)^4+20 b^2 (1+m)^2 n^2+64 b^4 n^4}+\frac {4 b n x^{1+m} \cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2+16 b^2 n^2}\\ \end {align*}

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Mathematica [A]  time = 4.03, size = 312, normalized size = 1.17 \[ \frac {1}{8} x^{m+1} \left (-\frac {4 \sin (2 b n \log (x)) \left ((m+1) \sin \left (2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )-2 b n \cos \left (2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )}{4 b^2 n^2+m^2+2 m+1}+\frac {4 \cos (2 b n \log (x)) \left ((m+1) \cos \left (2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+2 b n \sin \left (2 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )}{4 b^2 n^2+m^2+2 m+1}-\frac {\sin (4 b n \log (x)) \left ((m+1) \sin \left (4 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )-4 b n \cos \left (4 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )}{16 b^2 n^2+m^2+2 m+1}+\frac {\cos (4 b n \log (x)) \left ((m+1) \cos \left (4 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+4 b n \sin \left (4 \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )}{16 b^2 n^2+m^2+2 m+1}+\frac {3}{m+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*Cos[a + b*Log[c*x^n]]^4,x]

[Out]

(x^(1 + m)*(3/(1 + m) - (4*Sin[2*b*n*Log[x]]*(-2*b*n*Cos[2*(a - b*n*Log[x] + b*Log[c*x^n])] + (1 + m)*Sin[2*(a
- b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 4*b^2*n^2) + (4*Cos[2*b*n*Log[x]]*((1 + m)*Cos[2*(a - b*n*Lo
g[x] + b*Log[c*x^n])] + 2*b*n*Sin[2*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 4*b^2*n^2) - (Sin[4*b*
n*Log[x]]*(-4*b*n*Cos[4*(a - b*n*Log[x] + b*Log[c*x^n])] + (1 + m)*Sin[4*(a - b*n*Log[x] + b*Log[c*x^n])]))/(1
+ 2*m + m^2 + 16*b^2*n^2) + (Cos[4*b*n*Log[x]]*((1 + m)*Cos[4*(a - b*n*Log[x] + b*Log[c*x^n])] + 4*b*n*Sin[4*
(a - b*n*Log[x] + b*Log[c*x^n])]))/(1 + 2*m + m^2 + 16*b^2*n^2)))/8

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fricas [A]  time = 1.01, size = 273, normalized size = 1.03 \[ \frac {4 \, {\left (6 \, {\left (b^{3} m + b^{3}\right )} n^{3} x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + {\left (4 \, {\left (b^{3} m + b^{3}\right )} n^{3} + {\left (b m^{3} + 3 \, b m^{2} + 3 \, b m + b\right )} n\right )} x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{3}\right )} x^{m} \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + {\left (24 \, b^{4} n^{4} x + 12 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} + {\left (m^{4} + 4 \, m^{3} + 4 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, m + 1\right )} x \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{4}\right )} x^{m}}{m^{5} + 64 \, {\left (b^{4} m + b^{4}\right )} n^{4} + 5 \, m^{4} + 10 \, m^{3} + 20 \, {\left (b^{2} m^{3} + 3 \, b^{2} m^{2} + 3 \, b^{2} m + b^{2}\right )} n^{2} + 10 \, m^{2} + 5 \, m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*cos(a+b*log(c*x^n))^4,x, algorithm="fricas")

[Out]

(4*(6*(b^3*m + b^3)*n^3*x*cos(b*n*log(x) + b*log(c) + a) + (4*(b^3*m + b^3)*n^3 + (b*m^3 + 3*b*m^2 + 3*b*m + b
)*n)*x*cos(b*n*log(x) + b*log(c) + a)^3)*x^m*sin(b*n*log(x) + b*log(c) + a) + (24*b^4*n^4*x + 12*(b^2*m^2 + 2*
b^2*m + b^2)*n^2*x*cos(b*n*log(x) + b*log(c) + a)^2 + (m^4 + 4*m^3 + 4*(b^2*m^2 + 2*b^2*m + b^2)*n^2 + 6*m^2 +
4*m + 1)*x*cos(b*n*log(x) + b*log(c) + a)^4)*x^m)/(m^5 + 64*(b^4*m + b^4)*n^4 + 5*m^4 + 10*m^3 + 20*(b^2*m^3
+ 3*b^2*m^2 + 3*b^2*m + b^2)*n^2 + 10*m^2 + 5*m + 1)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*cos(a+b*log(c*x^n))^4,x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\cos ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*cos(a+b*ln(c*x^n))^4,x)

[Out]

int(x^m*cos(a+b*ln(c*x^n))^4,x)

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maxima [B]  time = 0.62, size = 3537, normalized size = 13.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*cos(a+b*log(c*x^n))^4,x, algorithm="maxima")

[Out]

1/16*(((cos(8*b*log(c))*cos(4*b*log(c)) + sin(8*b*log(c))*sin(4*b*log(c)) + cos(4*b*log(c)))*m^4 + 4*(cos(8*b*
log(c))*cos(4*b*log(c)) + sin(8*b*log(c))*sin(4*b*log(c)) + cos(4*b*log(c)))*m^3 + 16*(b^3*cos(4*b*log(c))*sin
(8*b*log(c)) - b^3*cos(8*b*log(c))*sin(4*b*log(c)) + b^3*sin(4*b*log(c)) + (b^3*cos(4*b*log(c))*sin(8*b*log(c)
) - b^3*cos(8*b*log(c))*sin(4*b*log(c)) + b^3*sin(4*b*log(c)))*m)*n^3 + 6*(cos(8*b*log(c))*cos(4*b*log(c)) + s
in(8*b*log(c))*sin(4*b*log(c)) + cos(4*b*log(c)))*m^2 + 4*(b^2*cos(8*b*log(c))*cos(4*b*log(c)) + b^2*sin(8*b*l
og(c))*sin(4*b*log(c)) + (b^2*cos(8*b*log(c))*cos(4*b*log(c)) + b^2*sin(8*b*log(c))*sin(4*b*log(c)) + b^2*cos(
4*b*log(c)))*m^2 + b^2*cos(4*b*log(c)) + 2*(b^2*cos(8*b*log(c))*cos(4*b*log(c)) + b^2*sin(8*b*log(c))*sin(4*b*
log(c)) + b^2*cos(4*b*log(c)))*m)*n^2 + 4*(cos(8*b*log(c))*cos(4*b*log(c)) + sin(8*b*log(c))*sin(4*b*log(c)) +
cos(4*b*log(c)))*m + 4*((b*cos(4*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(4*b*log(c)) + b*sin(4*b*lo
g(c)))*m^3 + 3*(b*cos(4*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(4*b*log(c)) + b*sin(4*b*log(c)))*m^2
+ b*cos(4*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(4*b*log(c)) + 3*(b*cos(4*b*log(c))*sin(8*b*log(c)
) - b*cos(8*b*log(c))*sin(4*b*log(c)) + b*sin(4*b*log(c)))*m + b*sin(4*b*log(c)))*n + cos(8*b*log(c))*cos(4*b*
log(c)) + sin(8*b*log(c))*sin(4*b*log(c)) + cos(4*b*log(c)))*x*x^m*cos(4*b*log(x^n) + 4*a) + 4*((cos(6*b*log(c
))*cos(4*b*log(c)) + cos(4*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)) + sin(4*b*log(c))*sin(2
*b*log(c)))*m^4 + 4*(cos(6*b*log(c))*cos(4*b*log(c)) + cos(4*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(4
*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*m^3 + 32*(b^3*cos(4*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(
c))*sin(4*b*log(c)) + b^3*cos(2*b*log(c))*sin(4*b*log(c)) - b^3*cos(4*b*log(c))*sin(2*b*log(c)) + (b^3*cos(4*b
*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(4*b*log(c)) + b^3*cos(2*b*log(c))*sin(4*b*log(c)) - b^3*cos
(4*b*log(c))*sin(2*b*log(c)))*m)*n^3 + 6*(cos(6*b*log(c))*cos(4*b*log(c)) + cos(4*b*log(c))*cos(2*b*log(c)) +
sin(6*b*log(c))*sin(4*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*m^2 + 16*(b^2*cos(6*b*log(c))*cos(4*b*log(c
)) + b^2*cos(4*b*log(c))*cos(2*b*log(c)) + b^2*sin(6*b*log(c))*sin(4*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*l
og(c)) + (b^2*cos(6*b*log(c))*cos(4*b*log(c)) + b^2*cos(4*b*log(c))*cos(2*b*log(c)) + b^2*sin(6*b*log(c))*sin(
4*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*m^2 + 2*(b^2*cos(6*b*log(c))*cos(4*b*log(c)) + b^2*cos(4*b*
log(c))*cos(2*b*log(c)) + b^2*sin(6*b*log(c))*sin(4*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*m)*n^2 +
4*(cos(6*b*log(c))*cos(4*b*log(c)) + cos(4*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)) + sin(4
*b*log(c))*sin(2*b*log(c)))*m + 2*((b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)) + b*
cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*m^3 + 3*(b*cos(4*b*log(c))*sin(6*b*log(c)
) - b*cos(6*b*log(c))*sin(4*b*log(c)) + b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))
*m^2 + b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)) + b*cos(2*b*log(c))*sin(4*b*log(c
)) - b*cos(4*b*log(c))*sin(2*b*log(c)) + 3*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(
c)) + b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*m)*n + cos(6*b*log(c))*cos(4*b*lo
g(c)) + cos(4*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*x
*x^m*cos(2*b*log(x^n) + 2*a) - ((cos(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)) + sin(4*b*l
og(c)))*m^4 + 4*(cos(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)) + sin(4*b*log(c)))*m^3 - 16
*(b^3*cos(8*b*log(c))*cos(4*b*log(c)) + b^3*sin(8*b*log(c))*sin(4*b*log(c)) + b^3*cos(4*b*log(c)) + (b^3*cos(8
*b*log(c))*cos(4*b*log(c)) + b^3*sin(8*b*log(c))*sin(4*b*log(c)) + b^3*cos(4*b*log(c)))*m)*n^3 + 6*(cos(4*b*lo
g(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)) + sin(4*b*log(c)))*m^2 + 4*(b^2*cos(4*b*log(c))*sin(8*
b*log(c)) - b^2*cos(8*b*log(c))*sin(4*b*log(c)) + (b^2*cos(4*b*log(c))*sin(8*b*log(c)) - b^2*cos(8*b*log(c))*s
in(4*b*log(c)) + b^2*sin(4*b*log(c)))*m^2 + b^2*sin(4*b*log(c)) + 2*(b^2*cos(4*b*log(c))*sin(8*b*log(c)) - b^2
*cos(8*b*log(c))*sin(4*b*log(c)) + b^2*sin(4*b*log(c)))*m)*n^2 + 4*(cos(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*
log(c))*sin(4*b*log(c)) + sin(4*b*log(c)))*m - 4*((b*cos(8*b*log(c))*cos(4*b*log(c)) + b*sin(8*b*log(c))*sin(4
*b*log(c)) + b*cos(4*b*log(c)))*m^3 + 3*(b*cos(8*b*log(c))*cos(4*b*log(c)) + b*sin(8*b*log(c))*sin(4*b*log(c))
+ b*cos(4*b*log(c)))*m^2 + b*cos(8*b*log(c))*cos(4*b*log(c)) + b*sin(8*b*log(c))*sin(4*b*log(c)) + 3*(b*cos(8
*b*log(c))*cos(4*b*log(c)) + b*sin(8*b*log(c))*sin(4*b*log(c)) + b*cos(4*b*log(c)))*m + b*cos(4*b*log(c)))*n +
cos(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)) + sin(4*b*log(c)))*x*x^m*sin(4*b*log(x^n) +
4*a) - 4*((cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)) + cos(2*b*log(c))*sin(4*b*log(c)
) - cos(4*b*log(c))*sin(2*b*log(c)))*m^4 + 4*(cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)
) + cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*m^3 - 32*(b^3*cos(6*b*log(c))*cos(4*b*l
og(c)) + b^3*cos(4*b*log(c))*cos(2*b*log(c)) + b^3*sin(6*b*log(c))*sin(4*b*log(c)) + b^3*sin(4*b*log(c))*sin(2
*b*log(c)) + (b^3*cos(6*b*log(c))*cos(4*b*log(c)) + b^3*cos(4*b*log(c))*cos(2*b*log(c)) + b^3*sin(6*b*log(c))*
sin(4*b*log(c)) + b^3*sin(4*b*log(c))*sin(2*b*log(c)))*m)*n^3 + 6*(cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*l
og(c))*sin(4*b*log(c)) + cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*m^2 + 16*(b^2*cos(
4*b*log(c))*sin(6*b*log(c)) - b^2*cos(6*b*log(c))*sin(4*b*log(c)) + b^2*cos(2*b*log(c))*sin(4*b*log(c)) - b^2*
cos(4*b*log(c))*sin(2*b*log(c)) + (b^2*cos(4*b*log(c))*sin(6*b*log(c)) - b^2*cos(6*b*log(c))*sin(4*b*log(c)) +
b^2*cos(2*b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*sin(2*b*log(c)))*m^2 + 2*(b^2*cos(4*b*log(c))*sin(6
*b*log(c)) - b^2*cos(6*b*log(c))*sin(4*b*log(c)) + b^2*cos(2*b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*s
in(2*b*log(c)))*m)*n^2 + 4*(cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)) + cos(2*b*log(c)
)*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*m - 2*((b*cos(6*b*log(c))*cos(4*b*log(c)) + b*cos(4*b*log
(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*m^3 + 3*(b*cos(6
*b*log(c))*cos(4*b*log(c)) + b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)) + b*sin(4*b
*log(c))*sin(2*b*log(c)))*m^2 + b*cos(6*b*log(c))*cos(4*b*log(c)) + b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(
6*b*log(c))*sin(4*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)) + 3*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*cos
(4*b*log(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*m)*n + c
os(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)) + cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*l
og(c))*sin(2*b*log(c)))*x*x^m*sin(2*b*log(x^n) + 2*a) + 6*((cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*m^4 + 64*(b
^4*cos(4*b*log(c))^2 + b^4*sin(4*b*log(c))^2)*n^4 + 4*(cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*m^3 + 6*(cos(4*b
*log(c))^2 + sin(4*b*log(c))^2)*m^2 + 20*(b^2*cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2 + (b^2*cos(4*b*log(c))
^2 + b^2*sin(4*b*log(c))^2)*m^2 + 2*(b^2*cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*m)*n^2 + 4*(cos(4*b*log(c)
)^2 + sin(4*b*log(c))^2)*m + cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*x*x^m)/((cos(4*b*log(c))^2 + sin(4*b*log(c
))^2)*m^5 + 5*(cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*m^4 + 64*(b^4*cos(4*b*log(c))^2 + b^4*sin(4*b*log(c))^2
+ (b^4*cos(4*b*log(c))^2 + b^4*sin(4*b*log(c))^2)*m)*n^4 + 10*(cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*m^3 + 10
*(cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*m^2 + 20*((b^2*cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*m^3 + b^2*c
os(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2 + 3*(b^2*cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*m^2 + 3*(b^2*cos(
4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*m)*n^2 + 5*(cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*m + cos(4*b*log(c))^
2 + sin(4*b*log(c))^2)

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mupad [B]  time = 3.59, size = 152, normalized size = 0.57 \[ \frac {3\,x\,x^m}{8\,m+8}+\frac {x\,x^m\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}{4\,m+4+b\,n\,8{}\mathrm {i}}+\frac {x\,x^m\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}\,1{}\mathrm {i}}{m\,4{}\mathrm {i}+8\,b\,n+4{}\mathrm {i}}+\frac {x\,x^m\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}}{16\,m+16+b\,n\,64{}\mathrm {i}}+\frac {x\,x^m\,{\mathrm {e}}^{-a\,4{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}}\,1{}\mathrm {i}}{m\,16{}\mathrm {i}+64\,b\,n+16{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*cos(a + b*log(c*x^n))^4,x)

[Out]

(3*x*x^m)/(8*m + 8) + (x*x^m*exp(a*2i)*(c*x^n)^(b*2i))/(4*m + b*n*8i + 4) + (x*x^m*exp(-a*2i)/(c*x^n)^(b*2i)*1
i)/(m*4i + 8*b*n + 4i) + (x*x^m*exp(a*4i)*(c*x^n)^(b*4i))/(16*m + b*n*64i + 16) + (x*x^m*exp(-a*4i)/(c*x^n)^(b
*4i)*1i)/(m*16i + 64*b*n + 16i)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*cos(a+b*ln(c*x**n))**4,x)

[Out]

Timed out

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